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Gabor Phase Retrieval in the Generalized Paley–wiener Space

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Abstract

Gabor phase retrieval is the problem of recovering a signal from the absolute values of Gabor transform, which has turned into a very active research area. In the paper, we define a generalized Paley–Wiener space, which is obtained by applying the special affine Fourier transform to \(L^{p}([-B,B])\,(2\le p\le \infty )\) and provide the uniqueness of the Gabor phase retrieval in the generalized Paley–Wiener space. When the window function \(\psi \) is the inverse special affine Fourier transform of \(\phi =2^{\frac{1}{4}}e^{-\pi x^{2}}\), we show that every signal in the generalized Paley–Wiener space can be uniquely recovered, up to a global sign, from its the magnitudes of Gabor measurement sampled on a rectangular lattice. Recently, Grohs, Liehr and Wellershoff studied the Gabor phase retrieval problems in the conventional Paley–Wiener space. The paper extends their results to the generalized Paley–Wiener space. Since the generalized Paley–Wiener space includes a broader class of signals, our results expand the applied range of the Gabor phase retrieval.

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Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Authors and Affiliations

  1. Tianjin Key Laboratory of Quantum Optics and Intelligent Photonics, College of Science, Tianjin University of Technology, Tianjin, 300384, China

    Qingyue Zhang, Liping Wu, Zhenli Guo & Bei Liu

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  1. Qingyue Zhang
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  2. Liping Wu
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  3. Zhenli Guo
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Correspondence to Liping Wu.

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This work was supported partially by the National Natural Science Foundation of China (12071230, 11971348, 11201336 and 11401435) and the Natural Science Research Project of Higher Education of Tianjin (2018KJ148).

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Zhang, Q., Wu, L., Guo, Z. et al. Gabor Phase Retrieval in the Generalized Paley–wiener Space. Circuits Syst Signal Process 43, 470–494 (2024). https://doi.org/10.1007/s00034-023-02485-1

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